Applications of the Karhunen-Loéve Transform for Basis Generation in the Response Matrix Method. Richard Reed

Transcription

1 Applications of the Karhunen-Loéve Transform for Basis Generation in the Response Matrix Method by Richard Reed B.S., Kansas State University, 2011 A Thesis submitted in partial fulfillment of the requirements for the degree Master of Science Department of Mechanical and Nuclear Engineering College of Engineering Kansas State University Manhattan, KS 2015 Approved by: Major Professor Jeremy Roberts

2 Abstract A novel approach based on the Karhunen-Loéve Transform (KLT) is presented for treatment of the energy variable in response matrix methods, which are based on the partitioning of global domains into independent nodes linked by approximate boundary conditions. These conditions are defined using truncated expansions of nodal boundary fluxes in each phase-space variable (i.e., space, angle, and energy). There are several ways in which to represent the dependence on these variables, each of which results in a trade-off between accuracy and speed. This work provides a method to expand in energy that can reduce the number of energy degrees of freedom needed for sub-0.1% errors in nodal fission densities by up to an of magnitude. The Karhunen-Loéve Transform is used to generate basis sets for expansion in the energy variable that maximize the amount of physics captured by low- moments, thus permitting low- expansions with less error than basis sets previously studied, e.g., the Discrete Legendre Polynomials (DLP) or modified DLPs. To test these basis functions, two 1-D test problems were developed: (1) a 10-pin representation of the junction between two heterogeneous fuel assemblies, and (2) a 70-pin representation of a boiling water reactor. Each of these problems utilized two cross-section libraries based on a 44-group and 238-group structure. Furthermore, a 2-D test problem based on the C5G7 benchmark is used to show applicability to higher dimensions.

3 Table of Contents List of Figures List of Tables List of Abbreviations List of Symbols List of Symbols Acknowledgements Dedication vi xi xii xiii xiii xiv xv 1 Introduction and Background Motivation Eigenvalue Response Matrix Method Overview Mathematical Formulation of ERMM Projection onto a Space and Angle Subspace Projection onto an Energy Group Subspace Response Matrix Formalism Objectives Summary of Previous Work Expanding in Energy Discrete Legendre Polynomials Modified Discrete Legendre Polynomials Summary Karhunen-Loève Transform 23 4 Test Problems and Models Test Problems D Test Problems D Test Problems Snapshot Generating Models iii

4 4.2.1 Snapshots for 10-Pin Test Problem Snapshots for BWR Test Problem Snapshots for C5G7 Test Problem SERMENT and DETRAN Results for 1-D Studies mdlp Comparison Energy Spectra for the Test Problems Pin Test Problem Group Results Group Results BWR Test Problem Configuration Group Results Group Results Configuration Group Results Group Results Configuration Group Results Group Results Higher-Order Angular Moments pin Problem Group Results Group Results BWR Test Problem Configuration Group Results Group Results Configuration Group Results Group Results Configuration Group Results Group Results Conclusion Results for 2-D Studies Energy Spectra for the Test Problem Nodal Fission Density Results Pin Power Results Conclusion Conclusions and Future Work Summary iv

5 7.2 Future Work Bibliography 89 A Parametric Studies 93 A.1 Snapshot Parameters A.1.1 Sensitivity to Snapshot Selection A.1.2 Sensitivity to Fineness of Spatial Mesh A.2 Database Parameters A.2.1 Sensitivity to Number of k Values in Databases A.2.2 Sensitivity to Range of k Values in Databases v

6 List of Figures 1.1 Example of 1-D ERMM node connections. The outgoing conditions J for a given node are the incident conditions on the adjacent node. Each node is computed individually, and the global solution is found by sweeping across all nodes Discrete Legendre Polynomials shown through 3 rd using 40 points Using 40 discrete points, cos(x) x [0, 2π] expanded by Discrete Legendre Polynomials to 4 th Example shape vector for creating mdlp basis functions Example basis functions for mdlp-1 shown through 3 rd Example basis functions for mdlp-2 shown through 3 rd Relative error in the L 2 norm as a function of expansion for various basis sets Basis functions for the KLT applied to an example problem Relative error in the L 2 norm as a function of expansion for KLT compared to other basis sets Configuration for 10-pin Test Problem Configuration for BWR Test Problem Configuration for Full-Core. Each square represents the area of a pin assembly Configuration for UO 2 fuel bundle. The green represents a UO 2 pincell, while the blue represents a guide tube modeled as a pincell filled with moderator Configuration for MOX bundle. The light red represents 4.3% MOX fuel, the medium red represents 7.0 % MOX fuel, and the dark red represents 8.7% MOX fuel. The blue represents moderator (i.e., light water) Configuration for pincell. The circular fuel element had a radius of 0.54 cm and was homogenized with cladding for this model Configuration for pincell junction; all Reflect BC; Every combination is used Configuration for small UO 2 fuel bundle. The green represents a UO 2 pincell, while the blue represents a guide tube modeled as a pincell filled with moderator vi

7 4.9 Configuration for small MOX bundle. The light red represents 4.3% MOX fuel, the medium red represents 7.0 % MOX fuel, and the dark red represents 8.7% MOX fuel. The blue represents moderator, light water Configuration for 1D approximation to the C5G7 benchmark. The green represents UO 2, the light red represents 4.3% MOX fuel, the medium red represents 7.0 % MOX fuel, and the dark red represents 8.7% MOX fuel, and the blue represents moderator Comparison of mdlp applied to the 10-pin problem using 44-group cross-section library Comparison of mdlp applied to the 10-pin problem using the 238-group cross-section library Flux spectrum for the 10-pin problem using 44-group cross-section library Flux spectrum for the 10-pin problem using 238-group cross-section library Flux spectrum for the BWR-Core 0 problem using 238-group crosssection library Flux spectrum for the BWR-Core 1 problem using 238-group crosssection library Flux spectrum for the BWR-Core 2 problem using 238-group crosssection library Performance of the KLT when applied to the 10-pin test problem with snapshots of only φ Performance of the KLT when applied to the 10-pin test problem with snapshots of only J left Performance of the KLT when applied to the 10-pin test problem with snapshots of both φ and leftward partial current Relative error for 238-group, 10-pin test problem using snapshots of only φ Relative error for 238-group, 10-pin test problem using snapshots of only J left Relative error for 238-group, 10-pin test problem using snapshots of both φ and J left Relative error for the 44-group, BWR-Core 0 test problem using snapshots of only φ Relative error for the 44-group, BWR-Core 0 test problem using snapshots of only J left Relative error for the 44-group, BWR-Core 0 test problem using snapshots of both φ and J left Relative error for 238-group, BWR-Core 0 test problem using snapshots of only φ Relative error for 238-group, BWR-Core 0 test problem using snapshots of only J left vii

8 5.19 Relative error for 238-group, BWR-Core 0 test problem using snapshots of both φ and J left Relative error for the 44-group, BWR-Core 1 test problem using snapshots of only φ Relative error for the 44-group, BWR-Core 1 test problem using snapshots of only J left Relative error for the 44-group, BWR-Core 1 test problem using snapshots of both φ and J left Relative error for 238-group, BWR-Core 1 test problem using snapshots of only φ Relative error for 238-group, BWR-Core 1 test problem using snapshots of only J left Relative error for 238-group, BWR-Core 1 test problem using snapshots of both φ and J left Relative error for the 44-group, BWR-Core 2 test problem using snapshots of only φ Relative error for the 44-group, BWR-Core 2 test problem using snapshots of only J left Relative error for the 44-group, BWR-Core 2 test problem using snapshots of both φ and J left Relative error for 238-group, BWR-Core 2 test problem using snapshots of only φ Relative error for 238-group, BWR-Core 2 test problem using snapshots of only J left Relative error for 238-group, BWR-Core 2 test problem using snapshots of both φ and J left Relative error for 44-group, 10-pin test problem using snapshots from the 10-pin model. Sets of snapshots are used separately for basis generation Relative error for 44-group, 10-pin test problem using snapshots from the 10-pin model. Sets of snapshots are combined together for basis generation Relative error for 238-group, 10-pin test problem using snapshots from the 10-pin model. Sets of snapshots are used separately for basis generation Relative error for 238-group, 10-pin test problem using snapshots from the 10-pin model. Sets of snapshots are combined together for basis generation Relative error for 44-group, BWR-Core 0 test problem using snapshots from the Full-Core model. Sets of snapshots are used separately for basis generation Relative error for 44-group, BWR-Core 0 test problem using snapshots from the Full-Core model. Sets of snapshots are combined together for basis generation viii

9 5.38 Relative error for 238-group, BWR-Core 0 test problem using snapshots from the Full-Core model. Sets of snapshots are used separately for basis generation Relative error for 238-group, BWR-Core 0 test problem using snapshots from the Full-Core model. Sets of snapshots are combined together for basis generation Relative error for 44-group, BWR-Core 1 test problem using snapshots from the Full-Core model. Sets of snapshots are used separately for basis generation Relative error for 44-group, BWR-Core 1 test problem using snapshots from the Full-Core model. Sets of snapshots are combined together for basis generation Relative error for 238-group, BWR-Core 1 test problem using snapshots from the Full-Core model. Sets of snapshots are used separately for basis generation Relative error for 238-group, BWR-Core 1 test problem using snapshots from the Full-Core model. Sets of snapshots are combined together for basis generation Relative error for 44-group, BWR-Core 2 test problem using snapshots from the Full-Core model. Sets of snapshots are used separately for basis generation Relative error for 44-group, BWR-Core 2 test problem using snapshots from the Full-Core model. Sets of snapshots are combined together for basis generation Relative error for 238-group, BWR-Core 2 test problem using snapshots from the Full-Core model. Sets of snapshots are used separately for basis generation Relative error for 238-group, BWR-Core 2 test problem using snapshots from the Full-Core model. Sets of snapshots are combined together for basis generation Flux spectrum for the C5G7 problem using 44-group cross-section library Relative error in fission density for 44-group, C5G7 test problem using snapshot of only φ Relative error in fission density for 44-group, C5G7 test problem using snapshot of J up and J down Relative error in fission density for 44-group, C5G7 test problem using snapshot of φ, J up, and J down Relative error in pin power for 44-group, C5G7 test problem using snapshot of only φ Relative error in pin power for 44-group, C5G7 test problem using snapshot of both J up and J down Relative error in pin power for 44-group, C5G7 test problem using snapshot of both φ, J up, and J down ix

10 6.8 Pin power heat map for the DETRAN reference solution. The upper left corner is the center of the core Error of pin powers in the SERMENT reference solution relative to the DETRAN reference solution. The upper left corner is the center of the core Error in the pin powers of the best performing KLT case (9th, Reduced Small-Core, snapshots of φ) relative to the SERMENT reference solution. The upper left corner is the center of the core Error in the pin powers of the best performing KLT case (9th, Reduced Small-Core, snapshots of J up and J down ) relative to the SERMENT reference solution. The upper left corner is the center of the core Error in the pin powers of the best performing KLT case (9th, Reduced Small-Core, snapshots of φ, J up, and J down ) relative to the SERMENT reference solution. The upper left corner is the center of the core A.1 Relative error for 238-group, BWR test problem when using various snapshot selection schemes. The schemes converge as the number of snapshots increases. The legend numbers correspond to the number of snapshots selected for KLT basis generation A.2 Relative error for 44-group, 10-pin test problem using snapshots from a spatially reduced model. The legend numbers correspond to the total number of available snapshots. The problem was reduced from 280 total snapshots A.3 Relative error for 238-group, BWR test problem a database of responses filled with a number of k values as indicated by the legend. The values spanned the range of ±0.15 of the the true value of k. The number of k values had no effect of the results A.4 Relative error for 238-group, BWR test problem a database of responses filled with eight k values. The values spanned the range of ± the value in the legend of the the true value of k. The size of the range no effect of the results x

11 List of Tables 4.1 Summary snapshot models for 10-pin Test Problem Summary of snapshot models for BWR Test Problem Summary of snapshot models for C5G7 test Problem xi

12 List of Abbreviations ERMM RMM DLP mdlp KLT POD PCA SVD BWR PWR MOX SERMENT DETRAN Eigenvalue Response Matrix Method Response Matrix Method Discrete Legendre Polynomials modified Discrete Legendre Polynomials Karhunen-Loève Transform Proper-Orthogonal Decomposition Principal Component Analysis Singular Value Decomposition Boiling Water Reactor Pressurized Water Reactor Mixed OXide fuel Solving Eigenvalue Response Matrix Equations using Nonlinear Techniques DETerministic TRANsport xii

13 List of Symbols del operator r position vector cm g energy group k eigenvalue T operator for transport processes F operator for neutron generation J +/ Current density in +/ direction cm 1 s 1 j angular current cm 1 s 1 sr 1 P m δ mn R n mth vector of basis set P Kronecker delta matrix of responses vector normal to surface ν neutrons produced per fission ρ phase-space vector Σ macroscopic cross section cm 1 φ scalar flux cm 1 s 1 χ PDF for fission emission energy ψ angular flux cm 1 s 1 sr 1 ˆΩ solid angle vector sr xiii

14 Acknowledgements I would take this time to thank just a few of those that have had a hand in shaping this work. First among those would be my advisor Professor Jeremy Roberts. His support and guidance has been instrumental in bettering myself and my work. In addition, I owe many thanks to Professor Larry Erickson, who has been a steadfast mentor and advisor throughout the years. I would like to thank Professors Ken Shultis and Hitesh Bindra for serving on my thesis committee and for always making time for any of my questions. I would also like to thank my colleagues, specifically Lyric, Kevin and Richard, for providing advice, encouragement, and much more. For the funding for this effort, I acknowledge the Kansas State University Nuclear Research Fellowship Program, which is generously sponsored by the U.S. Nuclear Regulatory Commission. xiv

15 Dedication Engineering is the art of modeling materials we do not wholly understand, into shapes we cannot precisely analyze so as to withstand forces we cannot properly assess, in such a way that the public has no reason to suspect the extent of our ignorance. A. R. Dyke xv

16 Chapter 1 Introduction and Background Nuclear reactors are extremely complicated systems, and modeling their dynamic behavior requires the solution of the neutron transport equation, a seven-dimensional (r, v, t) equation that describes the population of neutrons in the system. Unfortunately, to solve the complete neutron transport equation is all but impossible for realistic problems without first reducing the problem space in some way. For much of reactor analysis, it is reasonable to assume steady-state conditions, thus eliminating all time dependence from the problem. In addition, it is very difficult to account for the energy dependence directly, and a common approach is to use the multigroup method, in which the energy variable is discretized into G energy bins or groups. After these assumptions and the further assumption of isotropic scattering, the neutron transport equation reduces to ˆΩ ψ g (ˆΩ, r) + Σ t,g ψ g (ˆΩ, r) = [ 1 G Σ s,g g φ g (r) + χ g 4π k g =1 ] G νσ f,g φ g (r), g =1 (1.1) where ψ g represents the group-dependent angular flux and φ g is the group-dependent scalar flux. Furthermore, Σ t,g, Σ s,g g, and Σ f,g represent the group dependent cross sections for total, inscattering, and fission respectively. In addition, χ g is the fission spectrum, ν is the average number of neutrons emitted per fission, and the k-eigenvalue (or multiplication factor ) represents the balance of neutron gains 1

17 (by fission) to losses (by absorption and leakage). A reactor is critical, i.e., the neutron population is steady, when k = 1. Equation (1.1) may be used with any number of groups for which cross-section data is available. In any group structure, a system of G equations must be solved. Clearly, a greater number of groups will increase the difficulty of the problem, but it will also increase the accuracy, provided that the group structure is chosen wisely. Equation (1.1) may be cast in operator notation, i.e., T φ(ρ) = 1 Fφ(ρ), (1.2) k where T represents the transport processes, φ is the neutron flux, F represents the neutron generation, ρ represents the relevant phase-space, and k is the eigenvalue. A variety of approaches have been used to solve Eq. (1.2), which may broadly be categorized as deterministic (e.g., discrete ordinates) methods and stochastic (e.g., Monte Carlo) methods. This work has used a deterministic method exclusively, although much of the theory presented should also apply to stochastic methods. 1.1 Motivation To model a full reactor core with high-fidelity resolution in space, angle, and energy requires an enormous amount of computational power and memory. To illustrate the challenge, consider a typical pressurized water reactor (PWR) core with 193 assemblies, each with a array of fuel pins. Some reasonable parameters for resolving the localized pin powers are approximately 50 spatial cells in the x-y plane and 300 axial mesh points (i.e., 1 cm axial resolution) per pincell. Localized pin power resolution in energy requires approximately 100 groups, and similarly, to resolve the localized pin powers in angle requires approximately 100 angles. For discretization with just one unknown per cell, group, and angle, the total number 2

18 of unknowns is N =(193 assemblies) (17 2 pins ) (300 axial points) assembly (50 radial points) (100 groups) (100 angles) 2 (1.3) 1 4, for a single problem. For realistic analyses, a problem of this size would need to be solved repeatedly to account for thermal-hydraulic feedback effects or to model the change of material compositions over time. For such realistic problems, each spatial cell can be assumed to have a unique material, each of which is defined by O(100 2 ) floating point values. Hence, as shown, a large problem can quickly amass too many unknowns to solve in a reasonable amount of time on modern computers as storage requirements quickly rise above PByte levels. While the problems can be solved in principle, the solution requires too much time for production-level analyses, i.e., those for design of actual systems. Thus, the number of degrees of freedom must be reduced by reducing the spatial, angular, or energy resolution. There are, of course, repercussions for each type of reduction, but the goal is to minimize the effect of the reduction on the final solution of the neutron transport equation. If the problem is reduced spatially, we are no longer solving the same problem, while if energy or angle is reduced the fidelity of the model is reduced because some of the underlying physics is muted. In lattice physics, a complete solution is attained by solving the problem of interest several times to identify the dependence of each phase-space variable, then combine the separate solutions in such a way as to predict the global solution with some accuracy. It is common in lattice physics to first solve the problem with a 0-D representation in space and angle while continuous in energy to solve for the energy dependence. Next, a lattice physics solution will reduce the energy dependence to the multigroup approximation while using a 1-D or 2-D mesh in space along with high angular resolution to capture the angular dependence of the problem. These solutions are used to create approximate multigroup cross sections to be used in conjunction 3

19 with models with only a few energy groups, a course angular dependence, and a course 3-D spatial mesh. Many traditional methods for solving the neutron transport equation focus on a quickly computed solution by approximating the energy dependence, which simplifies the problem and in turn reduces the fidelity of the solution. However, in lattice physics, upwards of 200 energy groups or more are used to solve for a high-fidelity energy spectra. As such, a method that can bridge the gap between the quick, low-fidelity methods and the accurate, high-fidelity methods would be immensely valuable to the future of large-scale, reactor-physics simulations and analyses. Not only would the model be appropriately sized for modern computers, but it could also be more accurate than the current methods of whole core-analysis. 1.2 Eigenvalue Response Matrix Method Overview This work applies the eigenvalue response matrix method (ERMM). Response matrix methods are not new, and have been used in various formulations since the 1960 s, e.g., the work of Shimizu [1] and Shimizu et al. [2]. ERMM solves the reactor eigenvalue equation by decomposing the domain of a problem into independent nodes linked through approximate boundary conditions between each node. The boundary conditions are typically incident angular flux or current conditions. For ERMM, these conditions are represented by truncated, orthogonal basis expansions at the nodal boundaries. This approach effectively converts a large problem space into a large number of small, independent, transport problems, which creates many opportunities for parallelization of the algorithm. An example of ERMM is depicted in Fig. 1.1, where a sample problem is broken into M nodes. The mth node, where 1 m M, can be solved independently. In this way, the outgoing information(here denoted by J) from one cell becomes the incident conditions for the adjacent cell. The problem is then solved by assuming a 4

20 value for each of the initial incident boundary information. The cells are then solved for the outgoing information, which then becomes the new incident information for another solution. In this way, the method will continue until convergence criteria are met. J + J + J + J +... m-1 m m+1... J J J J Figure 1.1: Example of 1-D ERMM node connections. The outgoing conditions J for a given node are the incident conditions on the adjacent node. Each node is computed individually, and the global solution is found by sweeping across all nodes. In this case, the outgoing information is a boundary flux or current. The key to response matrix methods is that the outgoing conditions of a cell are not passed completely, but rather are projected onto a finite, orthogonal basis, and the coefficients of the resulting expansion become unknowns for the problem space. This projection reduces the number of degrees of freedom and, thus, reduces the size of the problem. An expansion is completed for each phase-space variable (i.e., space, angle, and energy), and the success of these expansions depends primarily on the selection of appropriate orthogonal bases for each phase-space variable. Response matrix methods are expensive in general, unless the problem space is greatly reduced in terms of the degrees of freedom as compared to more direct solutions. Thus, proper basis selection is paramount if ERMM is to compare in speed and accuracy to other methods for solving the transport equation. In short, basis sets that capture the detail of a high-fidelity transport solution with low- expansions are ideal for ERMM because fewer degrees of freedom are needed to achieve the desired accuracy, which reduces the problem space, and leads to an easier problem to solve computationally. This work builds on the previous effort presented by Roberts and Forget [3], which explored spatial and angular expansions in ERMM. This work extends their progress by utilizing a new, highly successful energy basis for use in ERMM based on the Karhunen-Loève transform, 5

21 which is discussed in detail in Chapter 3. These basis sets are applied to several illustrative problems, which provide insight to the success of the method. It is possible (and typical) to use different basis expansions for each variable; thus the expansion for each variable can be studied mostly independently of the other variables. However, there exists coupling between each phase-space variable. For example, the error caused by the expansion in energy is influenced by the expansion in space. This work used reference cases designed to isolate the effect of only one variable to the best extent possible. These reference cases are described in Chapter Mathematical Formulation of ERMM The following is a presentation of the time-independent, eigenvalue response matrix method. Although the following method is most closely linked to the work of Roberts and Forget [3], a presentation akin to that of Roberts [4] has been adapted for this work. For time-independent, multigroup problems, Eq. (1.2) may be rewritten as T ψ global (r, Ω, g) = 1 k Fψglobal (r, Ω, g), (1.4) where ρ has been replaced by the spatial coordinate r, the direction of travel Ω, and the energy group g. The global superscript in Eq. (1.4) indicates that the equation defines balance for the full problem of interest, and thus is the complete or global solution. ERMM works by breaking the global problem into small, independent or local nodes. To do so, let the global volume V be decomposed into I disjoint, convex, nodal sub-volumes V i that satisfy V = V 1 V2 VI. Furthermore, let the surface V i of V i be composed of S disjoint, planar surfaces V is that satisfy V i = V i1 ViS. While unnecessary, the condition of planar surfaces permits a compact notation. Finally, let r i and r is be shorthand for the variable r confined to values r V i and r V is, respectively. 6

22 Thus, the equivalent local transport equation for the ith node is T ψ local (r i, Ω, g) = 1 k Fψlocal (r i, Ω, g), (1.5) subject to S incident-flux boundary conditions ψ local (r is, Ω, g) = ψ global (r is, Ω, g), Ω n is < 0, (1.6) where the unit vector n is is the outward normal of surface V is. Equations (1.5) and (1.6) can be combined by casting incident-flux conditions as external sources such as, (T 1k F ) ψ local (r i, Ω, g) = j global (r i, Ω, g)δ(r i n is ), (1.7) where j = Ωψ is the angular current vector, and j is the magnitude of the angular current. For brevity, j is referred to as the angular current. This formulation allows the global problem to be formed by sweeping across each local node. The general solution of Eq. (1.7) for arbitrary incident conditions can be expressed as the convolution of the external source term with an appropriate kernel, or G S ψ(r i, Ω, g) = g =1 s =1 n is Ω <0 R f (r is, Ω, g r i, Ω, g)j(r is, Ω, g ) dω, (1.8) where G is the number of groups, and the local and global superscripts have been omitted. Similarly, exiting angular currents can be expressed as j + (r is, Ω, g) = G S g =1 s =1 n is Ω <0 R c (r is, Ω, g r is, Ω, g)j(r is, Ω, g ) dω, (1.9) where n is Ω > 0. The integration kernels R f and R c are flux- and currentresponse functions, which represent the angular flux and outgoing angular current at one point in phase-space due to a unit, incident current at another point in phase-space. However, some effort is required to convert Eqs. (1.8) and (1.9) into 7

23 a practical form. The goal is to reduce the effort needed to compute the global solution; thus, some approximations should be made for each phase-space variable Projection onto a Space and Angle Subspace The first approximations are the treatment of the angular and spatial dependence. Projection of local angular currents and fluxes onto a finite subspace represented by an orthogonal basis allows the use of a truncated basis to represent space and angle. Using a truncated basis reduces the problem space, and, hence, the computation cost of ERMM at the expense of reduced accuracy. Let a finite basis be constructed with a set of functions P m (r, Ω), m = 0, 1,... M, that are orthonormal over some domain of interest (i.e., a volume or a surface). Then the angular flux can be approximated as M ψ(r i, Ω, g) ψi m (g)pf m (r i, Ω), (1.10) m=0 where ψi m (g) = ψ(r i, Ω, g)pf m (r i, Ω) dωd 3 r i, (1.11) V i 4π and the f subscript denotes a basis suitable for the angular flux. Angular currents can similarly be approximated as j ± (r is, Ω, g) L l=0 j ±l is (g)p l c(r is, Ω), n is Ω 0, (1.12) where j ±l is (g) = V is n is Ω 0 j(r is, Ω, g)p l c(r is, Ω) dωd 2 r is, (1.13) and the c subscript denotes a basis suitable for the angular current defined on a boundary surface. Then, substitution of Eqs. (1.10) and (1.12) into Eq. (1.8) yields M ψi m (g)pf m (r i, Ω) m=0 G S L g =1 s =1 l =0 j l is (g ) R f, P l c, (1.14) 8

24 where variables have been suppressed and indicates the appropriate space and angle integration. Multiplication of Eq. (1.14) by P m f (r i, Ω) and integration of the result over space and angle leads to a set of flux moments defined by ψ m i (g) G S L g =1 s =1 l =0 j l is (g )R s l m fi (g g), (1.15) where R s l m fi (g g) R f, P l f, P m f. (1.16) Similarly, the outgoing angular currents can also be projected to yield the moments j +l is (g) G S L g =1 s =1 l =0 j l is (g )R s l sl ci (g g). (1.17) By choosing appropriate basis sets for expanding the spatial and angular variables, few terms are required for sufficient accuracy. For this work, Jacobi polynomials were used for the angular expansion, and Discrete Legendre Polynomials were used for the spatial expansion. The work did not focus on optimization of the spatial and angular basis functions, but rather the best cases from the work of Roberts and Forget [3] were used Projection onto an Energy Group Subspace This work focused on the basis sets for energy expansion, and the projection for energy is formulated similarly to the space-angle projection. This treatment will eliminate an explicit dependence on g. Because g is a discrete variable, bases used to represent dependence on g consist of discrete functions P h (g), h = 0, 1,..., H that satisfy G P h (g)p h (g) = δ hh, (1.18) g 9

25 where δ hh is the Kronecker-δ. By using such a discrete basis, group-dependent flux moments defined by Eq. (1.15) can be approximated as ψ m i (g) H h=0 ψ mh i P h (g), (1.19) where ψ mh i = G ψi m (g)p h (g). (1.20) g=1 Likewise, current moments defined by Eq. (1.17) can be approximated as j l is(g) H h=0 j lh is P h (g), (1.21) where j lh is = G ji(g)p l h (g). (1.22) g=1 Substitution of Eqs. (1.19) and (1.21) into Eq. (1.15) yields H h=0 ψi mh P h (g) G S L H g =1 s =1 l =0 h =0 j l h is (g ) R f, P l f, P m f. (1.23) Next, multiplication of the result by P h (g), and summation over energy leads to ψ mh i S L H s =1 l =0 h =0 j l h is R s l h mh fi, (1.24) where R s l h mh fi R f, P l f, P m f, P h, (1.25) and the outer brackets represent summation, not integration over g. Similarly, current moments are defined as j +lh is S L H s =1 l =0 h =0 j l h is R s l h slh ci, (1.26) where R s l h slh ci R c, P l c, P l c, P h. (1.27) 10

26 Computation of response function moments R s l h mh fi and R s l h slh ci requires evaluation of Eqs. (1.8) and (1.9) in which the incident current is equal to P l c (r is, Ω)P h (g) for each allowed combination of i, s, l, and h Response Matrix Formalism Equations (1.24) and (1.26) can be represented as nodal response matrix equations, i.e., ψ i = R fi j i j + i = R ci j i, (1.28) where ψ i and j ± i are vectors of nodal moments and R i s are matrices of nodal response function moments. Response matrix equations for the entire spatial domain can then be written as ψ = R f j j + = R c j. (1.29) By redirecting outgoing currents from one node as incident currents to another via j = Mj +, where the matrix M represents geometry and boundary conditions, the global equations become ψ = R f j j = MR c j. (1.30) In this formulation, the flux is dependent only on incident currents, and the response matrices R f and R c are functions of the k-eigenvalue. When k is not converged, the balance defined by Eq. (1.30) generally does not have a solution. As an alternative, the current equation can be rewritten as the nonlinear eigenvalue equation MR c (k)j = λj. (1.31) 11

27 After determining j by solving Eq. (1.31) for an assumed value of k, the flux and other volume moments can be determined. Subsequently, a new value for k is generated similarly to the traditional power method by using the standard balance relation of gains-to-losses. The new k is then used to find j, etc., until the solution has converged. A more detailed presentation of algorithms for solving the response matrix equations is given by Roberts and Forget [3]. 1.3 Objectives The primary focus of this work is to reduce by an of magnitude the number of energy degrees of freedom needed to achieve sub-0.1% error or better in the relative fission density error or pin powers. The fission density is directly related to the power levels throughout the problem. To evaluate the method, several test problems were devised to test the energy expansion including two 1-D problems and one 2-D problem. The project tested several facets of the orthogonal basis sets produced by the Karhunen-Loève Transform (KLT), which was used for expansion in energy. The KLT uses snapshots (discussed in detail in Chapter 4) to generate the basis sets. Thus, the work has considered several different models from which to generate snapshots. Each unique set of basis functions is used to expand the energy dependence for the test problem, and results are generated as the relative error in the fission density for the expansion as a function of. The results of the 1-D test problems are presented in Chapter 5, while the results of the 2-D test problem are presented in Chapter 6. During the course of the project, several parametric studies were developed to test the impact of different aspects of the models. These parametric studies are presented in Appendix A. 12

28 Chapter 2 Summary of Previous Work Response matrix methods have traditionally used a full multigroup representation of the energy variable, which is to say that there is no basis truncation for the energy basis. Response matrix methods have been used successfully with a variety of energy group structures, ranging from three to 190 groups [5 7]. One of the first studies of a truncated energy expansion for response matrix methods was the work of Roberts [8], who used Discrete Legendre Polynomials (DLP) and a modified version of DLP (mdlp) to expand in energy. His approach compared the basis expansions to a full multigroup approximation of the energy variable. This chapter aims to explore the previous work in expanding the energy variable, beginning with a brief overview of how the full multigroup approach can be represented in the formalism of Chapter 1, followed by a discussion of DLPs and problem-specific, modified DLPs. 2.1 Expanding in Energy In to represent the multigroup method exactly in ERMM, the energy variable is represented by a set of Kronecker-δ functions defined by P h δ (g) = δ h,g 1, g = 1, 2,..., G, (2.1) 13

29 where 1, if h = g 1, δ h,g 1 = 0, if h g 1. (2.2) When a complete set of these vectors (i.e., H = G 1) is used, a generic response function moment R s l h slh can be rewritten as R s l g slg. Thus, the group-to-group transfer process is represented in the traditional form. However, this approach requires that all energy groups be included, thus the number of energy degrees of freedom is equal to the number of groups. This method works well when the number of groups is low, but large models become prohibitively expensive when a detailed energy treatment (i.e., more than approximately 10 groups) is used. If a Kronecker set is truncated, a significant amount of the physics is lost, and the expansion has significantly reduced accuracy. Hence, the Kronecker set should not be used with energy reduction. In to improve ERMM performance, a basis set for energy is sought that will capture many-group fidelity while requiring many fewer degrees of freedom (H in Eqs. (1.24) and (1.26)) than a full multigroup analysis (a solution utilizing the complete set of Kronecker vectors). A way to reduce the energy degrees of freedom is to use an orthogonal basis to approximate the functionality of every group, thus converting the energy degrees of freedom for the problem into the coefficients of expansion. In this case, the energy degrees of freedom are equal in number to G, because H + 1 basis functions are required to expand exactly a vector of length G. The problem is then simplified by using a lower expansion than G. If the low- vectors in a basis set can better approximate the energy dependence, then the truncation should minimize effect of the error introduced such an expansion. The previous work by Roberts [8] suggested that incorporating physics directly into the basis functions may be more efficient than standard basis sets, and can lead to accuracy close to that of a full multigroup treatment using G groups without needing H + 1 energy degrees of freedom. 14

30 Each of the basis sets presented here are computed in orthonormal form. This formulation leads to quick determination of the expansion coefficients similarly to Eqs. (1.20) and (1.22), rewritten as a i = G w(g)p i (g)f(g), (2.3) g=0 where a i is the ith coefficient of expansion, w(g) is the weight associated with the basis function, P i (g) is the ith basis function, and f(g) is the function to expand. The reconstructed function is given by f(g) I a i P i (g) (2.4) i=0 where I is the expansion. Equation (2.4) is defined akin to Eqs. (1.19) and (1.21). Because a reduced number of energy degrees of freedom is desired, a basis that incorporates some physics of the problem is likely to provide the best expansion. The basis that includes physics should then be compared to more standard basis sets, such as the Discrete Legendre Polynomials (DLPs). 2.2 Discrete Legendre Polynomials The Legendre polynomials form a standard and proven basis that is used throughout computational physics. The discrete versions of the Legendre polynomials are shown in Fig When using the DLPs, the functions are represented by vectors of length equal to the number of discretized points. The vectors shown in Fig. 2.1 have been orthonormalized over the range of the vectors. Recent work investigated the use of discrete Legendre polynomials (DLPs) for expansion in energy [8 10]. The set of polynomials is generated using the Gram-Schmidt process to orthogonalize the discrete monomials M h (g) = g h, g = 0, 1,..., G 1. To illustrate, let G = 5, 15

31 Normalized Basis Function x P 0 P 1 P 2 P 3 Figure 2.1: Discrete Legendre Polynomials shown through 3 rd using 40 points for which the zeroth- DLP vector is defined as PDLP(:) 0 M 0 (:) = G g=1 M 0 (g) 5 = 5 [1, 1, 1, 1, 1]. (2.5) To define the first- DLP vector, let P DLP(:) 1 = M 1 (:) leading to ( G ) PDLP(g)M 0 1 (g) PDLP(:) 0, (2.6) g=1 P 1 DLP(:) = = G P 1 DLP (:) P g=1 DLP 1 (g) P DLP 1 (g) 10 5 [ 2, 1, 0, 1, 2]. (2.7) This procedure can be repeated for DLP vectors of arbitrary, up through one less than number of energy groups used. For the provided example, the second-, 16

32 third-, and fourth- DLP vectors are defined as P 2 DLP(:) = P 3 DLP(:) = P 4 DLP(:) = 146 [8, 1, 4, 1, 8] [ 16, 7, 0, 7, 16] [80, 85, 0, 85, 80]. (2.8) The DLPs can be use to truncate an expansion at any desired with varying levels of accuracy, e.g., use Eq. (2.4) except let I take a value less than the number of values in the expanded function, f i. To illustrate, let f(x) be defined as f(x) = cos(x) 0 x 2π. (2.9) To expand f(x) with the set of DLPs, F is first discretized by evaluating it at a number of points, e.g., 40 points. The expansion coefficients are then calculated by Eq. (2.3), and are then used to approximate f to various s using Eq. (2.4). The results of such expansions are given in Fig. 2.2, where the approximate function is plotted alongside the discretized f(x). Only the even s are shown in the figure because cos(x) is an even function, and hence all expansion coefficients for the odd functions of the DLPs are equal to zero. It is evident from Fig. 2.2 that increasing the expansion reduces the error of the approximation because more terms have been retained. However, adding odd functions to the expansion does not reduce the error of the approximation. In general, an expansion will converge as the basis set becomes more complete (where completeness is determined by the range of the basis set in the vector space of the function to expand), but it is not guaranteed that the expansion will converge to f(x). In the case of the neutron transport equation, the dependence of the solution on the expansions is non-linear in general; thus, the error cannot be guaranteed to decrease with increasing expansion, but the error is expected to decrease on the average as the is increased. 17

33 Function Value Function Order=0 Order=2 Order= x Figure 2.2: Using 40 discrete points, cos(x) x [0, 2π] expanded by Discrete Legendre Polynomials to 4 th 2.3 Modified Discrete Legendre Polynomials To improve the DLPs as a basis for energy expansion, the polynomials are first modified by superimposing a shape vector s on each basis vector, leading to the intermediate vectors P h mdlp(g) = P h DLP(g)s(g), g = 1, 2,..., G. (2.10) The modified Discrete Legendre Polynomial (mdlp) vectors P h mdlp are subsequently found by orthonormalizing the vectors P mdlp h. This formulation is referred to as Type 1 mdlp (mdlp-1) [8]. It is obvious that to expand a known function using the mdlp-1 basis, the best shape vector would be the function itself. The power of modifying the DLP vectors is observed when using the same basis set to expand several different but related functions, e.g., the scalar flux as a function of energy for different spatial cells in a test problem. As an example of mdlps, consider a 10-pin test problem consisting of 5 pins of UO 2 next to 5 pins of mixed oxide (MOX), i.e., plutonium-bearing, fuel with reflective boundary conditions (this example problem is the first 1-D test problem discussed 18

34 Shape Vector Energy Group Figure 2.3: Example shape vector for creating mdlp basis functions in Chapter 4 and is shown in Fig. 4.1). This problem can be discretized spatially into several regions, i.e., N spatial cells. Then the scalar flux φ can be calculated for each spatial cell, which leads to G group-dependent vectors that represent the scalar flux in each spatial cell. In this case, a shape vector representing the spatially-averaged scalar flux as a function of energy is used to modify the DLP vectors and form the mdlp basis. This example shape vector is shown in Fig This shape vector will create the basis functions shown in Fig. 2.4 when using mdlp-1. A second type of mdlp (mdlp-2) is formed by multiplying only the first DLP vector (i.e., the flat vector) with the shape function, then orthonormalizing the set of vectors [8]. A sample of mdlp-2 basis vectors is shown in Fig The efficiency of the mdlp vectors are shown by computing the error due to expansion as a function of. The error is computed as ɛ = H (f(h) f(h)) 2 H, (2.11) h=1 f(h)2 h=1 where f(h) is the hth element of the approximate function f as defined in Eq. (2.4). Equation (2.11) is the discrete definition of the L 2 norm, which measures the deviation between two vectors in a least squares sense. Figure 2.6 shows the 19

35 Normalized Basis Function P 0 P 1 P 2 P x Figure 2.4: Example basis functions for mdlp-1 shown through 3 rd Normalized Basis Function P 0 P 1 P 2 P x Figure 2.5: Example basis functions for mdlp-2 shown through 3 rd performance of both types of mdlp compared to DLP when used to expand the set of scalar flux vectors described for the example 10-pin case. Note that at the full ( = 43 for this example), all of the basis sets converge to within machine precision. It is clear that mdlp-1 outperforms mdlp-2, which suggests that more of the physics of the problem is captured by the lower- functions of mdlp-1. Thus, for this problem, the functions of mdlp-1 can better approximate 20